Question: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}5x+9y &= 2 \\ -6x-9y &= -8\end{align*}$
Begin by moving the $x$ -term in the second equation to the right side of the equation. $-9y = 6x-8$ Divide both sides by $-9$ to isolate $y$ $y = {-\dfrac{2}{3}x + \dfrac{8}{9}}$ Substitute this expression for $y$ in the first equation. $5x+9({-\dfrac{2}{3}x + \dfrac{8}{9}}) = 2$ $5x - 6x + 8 = 2$ Simplify by combining terms, then solve for $x$ $-1x + 8 = 2$ $-1x = -6$ $x = 6$ Substitute $6$ for $x$ back into the top equation. $5( 6)+9y = 2$ $30+9y = 2$ $9y = -28$ $y = -\dfrac{28}{9}$ The solution is $\enspace x = 6, \enspace y = -\dfrac{28}{9}$.